 15.2.15.1.51: If a candidate receives a majority of firstplace votes in an elect...
 15.2.15.1.52: If a candidate is favored when compared headtohead with every oth...
 15.2.15.1.53: A candidate who wins a first election then gains additional support...
 15.2.15.1.54: If a candidate is the winner of an election and in a second electio...
 15.2.15.1.55: A voting method that may not satisfy any of the fairness criteria i...
 15.2.15.1.56: A voting method that always satisfies the majority criterion but ma...
 15.2.15.1.57: A voting method that always satisfies the majority criterion and he...
 15.2.15.1.58: A voting method that always satisfies the majority criterion and th...
 15.2.15.1.59: Annual Meeting Members of the board of directors of the American Nu...
 15.2.15.1.60: Restructuring a Company The board of directors at The Limited is co...
 15.2.15.1.61: Hiring a New Director Irvine Valley College is hiring a new directo...
 15.2.15.1.62: Party Theme The children in Ms. Cohns seventhgrade class are votin...
 15.2.15.1.63: Residence Hall Improvements The administration at St. Cloud State U...
 15.2.15.1.64: Design of a Technology Building The planning committee of Tulsa Com...
 15.2.15.1.65: Preference for Grape Jelly Twentyone people are asked to taste tes...
 15.2.15.1.66: A Taste Test Twentyseven people are surveyed in the Mall of Americ...
 15.2.15.1.67: Plurality: Irrelevant Alternatives Criterion Suppose that the plura...
 15.2.15.1.68: Plurality: Irrelevant Alternatives Criterion Suppose that the plura...
 15.2.15.1.69: Borda Count: Irrelevant Alternatives Criterion Suppose that the Bor...
 15.2.15.1.70: Borda Count: Irrelevant Alternatives Criterion Suppose that the Bor...
 15.2.15.1.71: Plurality with Elimination: Monotonicity Criterion Suppose that the...
 15.2.15.1.72: Plurality with Elimination: Monotonicity Criterion Suppose that the...
 15.2.15.1.73: Pairwise Comparison Method: Monotonicity Criterion Suppose that the...
 15.2.15.1.74: Borda Count: Monotonicity Criterion Suppose that the Borda count me...
 15.2.15.1.75: Pairwise Comparison: Irrelevant Alternatives Criterion Suppose that...
 15.2.15.1.76: Pairwise Comparison: Irrelevant Alternatives Criterion Suppose that...
 15.2.15.1.77: Borda Count: Majority Criterion Suppose that the Borda count method...
 15.2.15.1.78: Borda Count: Majority Criterion Suppose that the Borda count method...
 15.2.15.1.79: Spring Trip The History Club of St. Louis is voting on which city t...
 15.2.15.1.80: Investment Choices The Motley Crew Investing Club wishes to purchas...
 15.2.15.1.81: Selecting a Spokesperson The Campbell Soup Company is selecting a n...
 15.2.15.1.82: Electing a Parks Director The City of Birmingham is electing a new ...
 15.2.15.1.83: Explain why the plurality method always satisfies the monotonicity ...
 15.2.15.1.84: Construct a preference table with four candidates and four rankings...
 15.2.15.1.85: Construct a preference table with three candidates and three rankin...
 15.2.15.1.86: Construct a preference table with three candidates and four ranking...
 15.2.15.1.87: Construct a preference table with four candidates and three ranking...
 15.2.15.1.88: Voting Strategy Consider the preference table below. Assume that a ...
 15.2.15.1.89: Choose a country other than the United States and write a research ...
 15.2.15.1.90: Write a research paper on the history of voting methods. Include ho...
Solutions for Chapter 15.2: Voting and Apportionment
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 15.2: Voting and Apportionment
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Chapter 15.2: Voting and Apportionment includes 40 full stepbystep solutions. Since 40 problems in chapter 15.2: Voting and Apportionment have been answered, more than 74079 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column space C (A) =
space of all combinations of the columns of A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).